Nonassociative Geometry: Friedmann–robertson–walker Spacetime

Abstract

In (Phys. Rev. D 62 (2000) 081501(R)) we proposed a unified description of continuum and discrete spacetime based on nonassociative geometry. It follows from our approach that at distances comparable with Planck length the standard concept of spacetime might be replaced by the nonassociative discrete structure, and nonassociativity is the algebraic equivalent of the curvature. In the framework of the nonassociative geometry we introduce discrete Friedmann-Robertson-Walker (FRW) model and show that the standard FRW spacetime appears as its 'continuum limit'.

How Nonassociative Geometry Describes a Discrete Spacetime

“…Since in case of discrete spaces we lack a smooth structure, we must use only algebraic structures to study them. Thus, it is quite natural to employ nonassociative geometry as the adequate algebraic approach (for details see [15][16][17][18]).…”

“…Since some information is lost during triangulation in the form of "missing" vertices (those that would have been closer than ℓ p ), the evolution is irreversible. Thus, in our approach the arrow of time, being related to the partial ordering, appears naturally [18].…”

“…This discreteness is motivated by several heuristic arguments, but so far has not been deduced rigorously. In Nesterov and Sabinin [15][16][17][18], we have proposed a new unified approach, based on nonassociative geometry, for describing both continuum and discrete spacetime. Among advanced models that propose discreteness, two that will be relevant to our work are Causal Sets (CS) [19][20][21] and Causal Dynamical Triangulations (CDT) [14,[22][23][24][25][26][27][28].…”

Gauging the twisted Poincaré symmetry as a noncommutative theory of gravitation

“…Since the introduction of a gauge covariant twist breaks the associativity of the algebra of functions on noncommutative space-time, both in the internal and external gauge symmetry cases, we may have to consider space-time geometries that are also nonassociative, not only noncommutative. Indeed, there exist in the literature works on constructing nonassociative theories with some desired properties (see, e.g., [41,42,43] and references therein).…”

Gauge field theories with covariant star-product

“…As the introduction of a gauge covariant twist, defined with θ µν constant, breaks the associativity of the algebra of functions on noncommutative space-time, both in the internal and external gauge symmetry cases, we may have to consider space-time geometries that are also non-associative, not only noncommutative. Indeed, there exist in the literature works on constructing non-associative theories with some desired properties (see, e.g., [23][24][25][26][27] and references therein). However, non-associativity introduces many difficulties in formulating gauge models and they are practically non-attractive.…”